Diffusion models now generate high-quality, diverse samples, with an increasing focus on more powerful models. Although ensembling is a well-known way to improve supervised models, its application to unconditional score-based diffusion models remains largely unexplored. In this work we investigate whether it provides tangible benefits for generative modelling. We find that while ensembling the scores generally improves the score-matching loss and model likelihood, it fails to consistently enhance perceptual quality metrics such as FID on image datasets. We confirm this observation across a breadth of aggregation rules using Deep Ensembles, Monte Carlo Dropout, on CIF AR-10 and FFHQ. We attempt to explain this discrepancy by investigating possible explanations, such as the link between score estimation and image quality. We also look into tabular data through random forests, and find that one aggregation strategy outperforms the others. Finally, we provide theoretical insights into the summing of score models, which shed light not only on ensembling but also on several model composition techniques (e.g.

Healthcare sector indices consolidate the economic health of pharmaceutical, biotechnology, and healthcare service firms. The short-term movements in these indices are closely intertwined with capital allocation decisions affecting research and development investment, drug availability, and long-term health outcomes. This research investigates whether historical open-high-low-close (OHLC) index data contain sufficient information for predicting the directional movement of the opening index on the subsequent trading day. The problem is formulated as a supervised classification task involving a one-step-ahead rolling window. A diverse feature set is constructed, comprising original prices, volatility-based technical indicators, and a novel class of nowcasting features derived from mutual OHLC ratios. The framework is evaluated on data from healthcare indices in the U.S. and Indian markets over a five-year period spanning multiple economic phases, including the COVID-19 pandemic. The results demonstrate robust predictive performance, with accuracy exceeding 0.8 and Matthews correlation coefficients above 0.6. Notably, the proposed nowcasting features have emerged as a key determinant of the market movement. We have employed the Shapley-based explainability paradigm to further elucidate the contribution of the features: outcomes reveal the dominant role of the nowcasting features, followed by a more moderate contribution of original prices. This research offers a societal utility: the proposed features and model for short-term forecasting of healthcare indices can reduce information asymmetry and support a more stable and equitable health economy.

Self-Organizing Maps provide topology-preserving projections of high-dimensional data and have been widely used for visualization, clustering, and vector quantization. In this work, we show that the activation pattern of a SOM - the squared distances to its prototypes - can be inverted to recover the exact input under mild geometric conditions. This follows from a classical fact in Euclidean distance geometry: a point in $D$ dimensions is uniquely determined by its distances to $D{+}1$ affinely independent references. We derive the corresponding linear system and characterize the conditions under which the inversion is well-posed. Building upon this mechanism, we introduce the Manifold-Aware Unified SOM Inversion and Control (MUSIC) update rule, which enables controlled, semantically meaningful trajectories in latent space. MUSIC modifies squared distances to selected prototypes while preserving others, resulting in a deterministic geometric flow aligned with the SOM's piecewise-linear structure. Tikhonov regularization stabilizes the update rule and ensures smooth motion on high-dimensional datasets. Unlike variational or probabilistic generative models, MUSIC does not rely on sampling, latent priors, or encoder-decoder architectures. If no perturbation is applied, inversion recovers the exact input; when a target cluster or prototype is specified, MUSIC produces coherent semantic variations while remaining on the data manifold. This leads to a new perspective on data augmentation and controllable latent exploration based solely on prototype geometry. We validate the approach using synthetic Gaussian mixtures, the MNIST and the Faces in the Wild dataset. Across all settings, MUSIC produces smooth, interpretable trajectories that reveal the underlying geometry of the learned manifold, illustrating the advantages of SOM-based inversion over unsupervised clustering.

Fairness and privacy are two vital pillars of trustworthy machine learning. Despite extensive research on these individual topics, the relationship between fairness and privacy has received significantly less attention. In this paper, we utilize the information-theoretic measure Chernoff Information to highlight the data-dependent nature of the relationship among the triad of fairness, privacy, and accuracy. We first define Noisy Chernoff Difference, a tool that allows us to analyze the relationship among the triad simultaneously. We then show that for synthetic data, this value behaves in 3 distinct ways (depending on the distribution of the data). We highlight the data distributions involved in these cases and explore their fairness and privacy implications. Additionally, we show that Noisy Chernoff Difference acts as a proxy for the steepness of the fairness-accuracy curves. Finally, we propose a method for estimating Chernoff Information on data from unknown distributions and utilize this framework to examine the triad dynamic on real datasets. This work builds towards a unified understanding of the fairness-privacy-accuracy relationship and highlights its data-dependent nature.

The Muon optimizer, a matrix-structured algorithm that leverages spectral orthogonalization of gradients, is a milestone in the pretraining of large language models. However, the underlying mechanisms of Muon -- particularly the role of gradient orthogonalization -- remain poorly understood, with very few works providing end-to-end analyses that rigorously explain its advantages in concrete applications. We take a step by studying the effectiveness of a simplified variant of Muon through two case studies: matrix factorization, and in-context learning of linear transformers. For both problems, we prove that simplified Muon converges linearly with iteration complexities independent of the relevant condition number, provably outperforming gradient descent and Adam. Our analysis reveals that the Muon dynamics decouple into a collection of independent scalar sequences in the spectral domain, each exhibiting similar convergence behavior. Our theory formalizes the preconditioning effect induced by spectral orthogonalization, offering insight into Muon's effectiveness in these matrix optimization problems and potentially beyond.

The increasing reliance on human preference feedback to judge AI-generated pseudo labels has created a pressing need for principled, budget-conscious data acquisition strategies. We address the crucial question of how to optimally allocate a fixed annotation budget between ground-truth labels and pairwise preferences in AI. Our solution, grounded in semi-parametric inference, casts the budget allocation problem as a monotone missing data framework. Building on this formulation, we introduce Preference-Calibrated Active Learning (PCAL), a novel method that learns the optimal data acquisition strategy and develops a statistically efficient estimator for functionals of the data distribution. Theoretically, we prove the asymptotic optimality of our PCAL estimator and establish a key robustness guarantee that ensures robust performance even with poorly estimated nuisance models. Our flexible framework applies to a general class of problems, by directly optimizing the estimator's variance instead of requiring a closed-form solution. This work provides a principled and statistically efficient approach for budget-constrained learning in modern AI. Simulations and real-data analysis demonstrate the practical benefits and superior performance of our proposed method.

In this paper, the solution to the empirical risk minimization problem with $f$-divergence regularization (ERM-$f$DR) is presented and conditions under which the solution also serves as the solution to the minimization of the expected empirical risk subject to an $f$-divergence constraint are established. The proposed approach extends applicability to a broader class of $f$-divergences than previously reported and yields theoretical results that recover previously known results. Additionally, the difference between the expected empirical risk of the ERM-$f$DR solution and that of its reference measure is characterized, providing insights into previously studied cases of $f$-divergences. A central contribution is the introduction of the normalization function, a mathematical object that is critical in both the dual formulation and practical computation of the ERM-$f$DR solution. This work presents an implicit characterization of the normalization function as a nonlinear ordinary differential equation (ODE), establishes its key properties, and subsequently leverages them to construct a numerical algorithm for approximating the normalization factor under mild assumptions. Further analysis demonstrates structural equivalences between ERM-$f$DR problems with different $f$-divergences via transformations of the empirical risk. Finally, the proposed algorithm is used to compute the training and test risks of ERM-$f$DR solutions under different $f$-divergence regularizers. This numerical example highlights the practical implications of choosing different functions $f$ in ERM-$f$DR problems.

Online continual learning (OCL) methods adapt to changing environments without forgetting past knowledge. Similarly, online time series forecasting (OTSF) is a real-world problem where data evolve in time and success depends on both rapid adaptation and long-term memory. Indeed, time-varying and regime-switching forecasting models have been extensively studied, offering a strong justification for the use of OCL in these settings. Building on recent work that applies OCL to OTSF, this paper aims to strengthen the theoretical and practical connections between time series methods and OCL. First, we reframe neural network optimization as a parameter filtering problem, showing that natural gradient descent is a score-driven method and proving its information-theoretic optimality. Then, we show that using a Student's t likelihood in addition to natural gradient induces a bounded update, which improves robustness to outliers. Finally, we introduce Natural Score-driven Replay (NatSR), which combines our robust optimizer with a replay buffer and a dynamic scale heuristic that improves fast adaptation at regime drifts. Empirical results demonstrate that NatSR achieves stronger forecasting performance than more complex state-of-the-art methods.

Estimating the unknown reward functions driving agents' behaviors is of central interest in inverse reinforcement learning and game theory. To tackle this problem, we develop a unified framework for reward function recovery in two-player zero-sum matrix games and Markov games with entropy regularization, where we aim to reconstruct the underlying reward functions given observed players' strategies and actions. This task is challenging due to the inherent ambiguity of inverse problems, the non-uniqueness of feasible rewards, and limited observational data coverage. To address these challenges, we establish the reward function's identifiability using the quantal response equilibrium (QRE) under linear assumptions. Building upon this theoretical foundation, we propose a novel algorithm to learn reward functions from observed actions. Our algorithm works in both static and dynamic settings and is adaptable to incorporate different methods, such as Maximum Likelihood Estimation (MLE). We provide strong theoretical guarantees for the reliability and sample efficiency of our algorithm. Further, we conduct extensive numerical studies to demonstrate the practical effectiveness of the proposed framework, offering new insights into decision-making in competitive environments.

Real-world tabular databases routinely combine continuous measurements and categorical records, yet missing entries are pervasive and can distort downstream analysis. We propose Statistical-Neural Interaction (SNI), an interpretable mixed-type imputation framework that couples correlation-derived statistical priors with neural feature attention through a Controllable-Prior Feature Attention (CPFA) module. CPFA learns head-wise prior-strength coefficients $\{λ_h\}$ that softly regularize attention toward the prior while allowing data-driven deviations when nonlinear patterns appear to be present in the data. Beyond imputation, SNI aggregates attention maps into a directed feature-dependency matrix that summarizes which variables the imputer relied on, without requiring post-hoc explainers. We evaluate SNI against six baselines (Mean/Mode, MICE, KNN, MissForest, GAIN, MIWAE) on six datasets spanning ICU monitoring, population surveys, socio-economic statistics, and engineering applications. Under MCAR/strict-MAR at 30\% missingness, SNI is generally competitive on continuous metrics but is often outperformed by accuracy-first baselines (MissForest, MIWAE) on categorical variables; in return, it provides intrinsic dependency diagnostics and explicit statistical-neural trade-off parameters. We additionally report MNAR stress tests (with a mask-aware variant) and discuss computational cost, limitations -- particularly for severely imbalanced categorical targets -- and deployment scenarios where interpretability may justify the trade-off.